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Time dilationFrom Wikipedia, the free encyclopedia
This article is about a concept in physics. For the concept in sociology, see time
displacement.
In the theory of relativity, time dilation is an observed difference of elapsed time between
two events as measured by observers either moving relative to each other or differently
situated from gravitational masses. An accurate clock at rest with respect to one observer
may be measured to tick at a different rate when compared to a second observer's own
equally accurate clocks. This effect arises not from technical aspects of the clocks nor from
the fact that signals need time to propagate, but from the nature of space-time itself.
Contents
[hide]
1 Overview
o 1.1 Relative velocity time dilation
o 1.2 Gravitational time dilation
o 1.3 Time dilation: special vs. general theories of relativity
2 Simple inference of time dilation due to relative velocity
3 Time dilation due to relative velocity symmetric between observers
o 3.1 Temporal coordinate systems and clock synchronization
4 Overview of formulae
o 4.1 Time dilation due to relative velocity
o 4.2 Time dilation due to gravitation and motion together
5 Experimental confirmation
o 5.1 Velocity time dilation tests
o 5.2 Gravitational time dilation tests
o 5.3 Velocity and gravitational time dilation combined-effect tests
o 5.4 Muon lifetime
6 Time dilation and space flight
o 6.1 Time dilation at constant acceleration
o 6.2 Spacetime geometry of velocity time dilation
7 See also
8 References
[edit]Overview
Time dilation can arise from:
1. the relative velocity of motion between two observers, or
2. the difference in their distance from a gravitational mass.
[edit]Relative velocity time dilation
When two observers are in relative uniform motion and uninfluenced by any gravitational
mass, the point of view of each will be that the other's (moving) clock is ticking at
a slower rate than the local clock. The faster the relative velocity, the greater the magnitude
of time dilation. This case is sometimes called special relativistic time dilation. It is often
interpreted as time "slowing down" for the other (moving) clock. But that is only true from
the physical point of view of the local observer, and of others at relative rest (i.e. in the local
observer's frame of reference). The point of view of the other observer will be that again the
local clock (this time the other clock) is correct and it is the distant moving one that is slow.
From a local perspective, time registered by clocks that are at rest with respect to the local
frame of reference (and far from any gravitational mass) always appears to pass at the
same rate.[1]
[edit]Gravitational time dilation
Main article: Gravitational time dilation
There is another case of time dilation, where both observers are differently situated in their
distance from a significant gravitational mass, such as (for terrestrial observers) the Earth or
the Sun. One may suppose for simplicity that the observers are at relative rest (which is not
the case of two observers both rotating with the Earth—an extra factor described below). In
the simplified case, the general theory of relativitydescribes how, for both observers, the
clock that is closer to the gravitational mass, i.e. deeper in its "gravity well", appears to go
slower than the clock that is more distant from the mass (or higher in altitude away from the
center of the gravitational mass). That does not mean that the two observers fully agree:
each still makes the local clock to be correct; the observer more distant from the mass
(higher in altitude) measures the other clock (closer to the mass, lower in altitude) to be
slower than the local correct rate, and the observer situated closer to the mass (lower in
altitude) measures the other clock (farther from the mass, higher in altitude) to be faster
than the local correct rate. They agree at least that the clock nearer the mass is slower in
rate and on the ratio of the difference.
[edit]Time dilation: special vs. general theories of relativity
In Albert Einstein's theories of relativity, time dilation in these two circumstances can be
summarized:
In special relativity (or, hypothetically far from all gravitational mass), clocks that are
moving with respect to an inertial system of observation are measured to be running
slower. This effect is described precisely by the Lorentz transformation.
In general relativity, clocks at lower potentials in a gravitational field—such as in closer
proximity to a planet—are found to be running slower. The articles on gravitational time
dilation and gravitational red shift give a more detailed discussion.
Special and general relativistic effects can combine, for example in some time-scale
applications mentioned below.
In special relativity, the time dilation effect is reciprocal: as observed from the point of view
of either of two clocks which are in motion with respect to each other, it will be the other
clock that is time dilated. (This presumes that the relative motion of both parties is uniform;
that is, they do not accelerate with respect to one another during the course of the
observations.)
In contrast, gravitational time dilation (as treated in general relativity) is not reciprocal: an
observer at the top of a tower will observe that clocks at ground level tick slower, and
observers on the ground will agree about the direction and the ratio of the difference. There
is not full agreement, as all the observers make their own local clocks out to be correct, but
the direction and ratio of gravitational time dilation is agreed by all observers, independent
of their altitude.
[edit]Simple inference of time dilation due to relative velocity
Observer at rest sees time 2L/c.
Observer moving parallel relative to setup, sees longer path, time > 2L/c, same speed c.
Time dilation can be inferred from the observed fact of the constancy of the speed of light in
all reference frames.[2][3][4][5]
This constancy of the speed of light means, counter to intuition, that speeds of material
objects and light are not additive. It is not possible to make the speed of light appear faster
by approaching at speed towards the material source that is emitting light. It is not possible
to make the speed of light appear slower by receding from the source at speed. From one
point of view, it is the implications of this unexpected constancy that take away from
constancies expected elsewhere.
Consider a simple clock consisting of two mirrors A and B, between which a light pulse is
bouncing. The separation of the mirrors is L and the clock ticks once each time it hits a
given mirror.
In the frame where the clock is at rest (diagram at right), the light pulse traces out a path of
length 2Land the period of the clock is 2L divided by the speed of light:
From the frame of reference of a moving observer traveling at the speed v (diagram at
lower right), the light pulse traces out a longer, angled path. The second postulate of
special relativity states that the speed of light is constant in all frames, which implies a
lengthening of the period of this clock from the moving observer's perspective. That is to
say, in a frame moving relative to the clock, the clock appears to be running more
slowly. Straightforward application of the Pythagorean theorem leads to the well-known
prediction of special relativity:
The total time for the light pulse to trace its path is given by
The length of the half path can be calculated as a function of known quantities as
Substituting D from this equation into the previous and solving for Δt' gives:
and thus, with the definition of Δt:
which expresses the fact that for the moving observer the period of the
clock is longer than in the frame of the clock itself.
[edit]Time dilation due to relative velocity symmetric between observers
Common sense would dictate that if time passage has slowed for a
moving object, the moving object would observe the external world to be
correspondingly "sped up". Counterintuitively, special relativity predicts
the opposite.
A similar oddity occurs in everyday life. If Sam sees Abigail at a distance
she appears small to him and at the same time Sam appears small to
Abigail. Being very familiar with the effects of perspective, we see no
mystery or a hint of a paradox in this situation.[6]
One is accustomed to the notion of relativity with respect to distance: the
distance from Los Angeles to New York is by convention the same as the
distance from New York to Los Angeles. On the other hand, when
speeds are considered, one thinks of an object as "actually" moving,
overlooking that its motion is always relative to something else — to the
stars, the ground or to oneself. If one object is moving with respect to
another, the latter is moving with respect to the former and with equal
relative speed.
In the special theory of relativity, a moving clock is found to be ticking
slowly with respect to the observer's clock. If Sam and Abigail are on
different trains in near-lightspeed relative motion, Sam measures (by all
methods of measurement) clocks on Abigail's train to be running slowly
and similarly, Abigail measures clocks on Sam's train to be running
slowly.
Note that in all such attempts to establish "synchronization" within the
reference system, the question of whether something happening at one
location is in fact happening simultaneously with something happening
elsewhere, is of key importance. Calculations are ultimately based on
determining which events are simultaneous. Furthermore, establishing
simultaneity of events separated in space necessarily requires
transmission of information between locations, which by itself is an
indication that the speed of light will enter the determination of
simultaneity.
It is a natural and legitimate question to ask how, in detail, special
relativity can be self-consistent if clock A is time-dilated with respect to
clock B and clock B is also time-dilated with respect to clock A. It is by
challenging the assumptions built into the common notion of simultaneity
that logical consistency can be restored. Simultaneity is a relationship
between an observer in a particular frame of reference and a set of
events. By analogy, left and right are accepted to vary with the position
of the observer, because they apply to a relationship. In a similar
vein, Plato explained that up and down describe a relationship to the
earth and one would not fall off at the antipodes.
Within the framework of the theory and its terminology there is a relativity
of simultaneity that affects how the specified events are aligned with
respect to each other by observers in relative motion. Because the pairs
of putatively simultaneous moments are identified differently by different
observers (as illustrated in the twin paradox article), each can treat the
other clock as being the slow one without relativity being self-
contradictory. This can be explained in many ways, some of which
follow.
[edit]Temporal coordinate systems and clock synchronization
In Relativity, temporal coordinate systems are set up using a procedure
for synchronizing clocks, discussed by Poincaré (1900) in relation to
Lorentz's local time (see relativity of simultaneity). It is now usually called
the Einstein synchronization procedure, since it appeared in his 1905
paper.
An observer with a clock sends a light signal out at time t1 according to
his clock. At a distant event, that light signal is reflected back, and arrives
back at the observer at time t2 according to his clock. Since the light
travels the same path at the same rate going both out and back for the
observer in this scenario, the coordinate time of the event of the light
signal being reflected for the observer tE is tE = (t1 + t2) / 2. In this way, a
single observer's clock can be used to define temporal coordinates which
are good anywhere in the universe.
Symmetric time dilation occurs with respect to temporal coordinate
systems set up in this manner. It is an effect where another clock is
being viewed as running slowly by an observer. Observers do not
consider their own clock time to be time-dilated, but may find that it is
observed to be time-dilated in another coordinate system.
[edit]Overview of formulae
[edit]Time dilation due to relative velocity
Lorentz factor as a function of speed (in natural units where c=1). Notice that for small speeds
(less than 0.1), γ is approximately 1
The formula for determining time dilation in special relativity is:
where Δt is the time interval between two co-local events (i.e.
happening at the same place) for an observer in some inertial frame
(e.g. ticks on his clock) – this is known as the proper time, Δt ' is the
time interval between those same events, as measured by another
observer, inertially moving with velocity v with respect to the former
observer, v is the relative velocity between the observer and the
moving clock, c is the speed of light, and
is the Lorentz factor. Thus the duration of the clock cycle of a
moving clock is found to be increased: it is measured to be
"running slow". The range of such variances in ordinary life,
where v ≪ c, even considering space travel, are not great
enough to produce easily detectable time dilation effects and
such vanishingly small effects can be safely ignored. It is only
when an object approaches speeds on the order of 30,000
km/s (1/10 the speed of light) that time dilation becomes
important.
Time dilation by the Lorentz factor was predicted by Joseph
Larmor(1897), at least for electrons orbiting a nucleus. Thus "...
individual electrons describe corresponding parts of their orbits in
times shorter for the [rest] system in the ratio : " (Larmor
1897). Time dilation of magnitude corresponding to this (Lorentz)
factor has been experimentally confirmed, as described below.
[edit]Time dilation due to gravitation and motion together
High accuracy time keeping, low earth orbit satellite tracking,
and pulsar timing are applications that require the consideration
of the combined effects of mass and motion in producing time
dilation. Practical examples include the International Atomic
Time standard and its relationship with the Barycentric
Coordinate Time standard used for interplanetary objects.
Relativistic time dilation effects for the solar system and the Earth
can be modeled very precisely by the Schwarzschild solution to
the Einstein field equations. In the Schwarzschild metric, the
interval dtE is given by:[7][8]
(1
)
where:
dtE is a small increment of proper time tE (an interval that could be recorded on an
atomic clock);
dtc is a small increment in the coordinate tc (coordinate time);
dx, dy and dz are small increments in the three coordinates x, y, z of the clock's
position; and
GMi/ri represents the sum of the Newtonian gravitational potentials due to the
masses in the neighborhood, based on their distances rifrom the clock. This
sum GMi/ri includes any tidal potentials, and is represented as U (using the positive
astronomical sign convention for gravitational potentials).
The coordinate velocity of the clock is
(2
)
The coordinate time tc is the time that
would be read on a hypothetical
"coordinate clock" situated infinitely far
from all gravitational masses (U=0), and
stationary in the system of coordinates
(v=0). The exact relation between the rate
of proper time and the rate of coordinate
time for a clock with a radial component of
velocity is:
(3
)
where:
is the radial velocity, and
U = GMi/ri is the Newtonian potential, equivalent to half of the escape velocity
squared.
The above equation is exact
under the assumptions of the
Schwarzschild solution.
[edit]Experimental confirmation
See also: Tests of special
relativity
Time dilation has been tested a
number of times. The routine
work carried on in particle
accelerators since the 1950s,
such as those atCERN, is a
continuously running test of the
time dilation of special
relativity. The specific
experiments include:
[edit]Velocity time dilation tests
Ives and Stilwell (1938,
1941). The stated purpose
of these experiments was
to verify the time dilation
effect, predicted by Lamor-
Lorentz ether theory, due to
motion through the ether
using Einstein's suggestion
that Doppler effect in canal
rays would provide a
suitable experiment. These
experiments measured
the Doppler shift of the
radiation emitted
from cathode rays, when
viewed from directly in front
and from directly behind.
The high and low
frequencies detected were
not the classical values
predicted.
and and
i.e. for sources with invariant frequencies The high and low
frequencies of the radiation from the moving sources were measured as
and
as deduced by Einstein (1905) from the Lorentz transformation, when the source is
running slow by the Lorentz factor.
Rossi and
Hall (1941)
compared the
population of
cosmic-ray-
produced muon
s at the top of a
mountain to that
observed at sea
level. Although
the travel time
for the muons
from the top of
the mountain to
the base is
several muon
half-lives, the
muon sample at
the base was
only moderately
reduced. This is
explained by the
time dilation
attributed to
their high speed
relative to the
experimenters.
That is to say,
the muons were
decaying about
10 times slower
than if they were
at rest with
respect to the
experimenters.
Hasselkamp,
Mondry, and
Scharmann[9] (1
979) measured
the Doppler shift
from a source
moving at right
angles to the
line of sight
(thetransverse
Doppler shift).
The most
general
relationship
between
frequencies of
the radiation
from the moving
sources is given
by:
as deduced by Einstein (1905)[1]. For ( ) this reduces
to fdetected = frestγ. Thus there is no transverse Doppler shift, and the lower frequency of
the moving source can be attributed to the time dilation effect alone.
In 2010
time
dilation
was
observe
d at
speeds
of less
than 10
meters
per
second
using
optical
atomic
clocks
connecte
d by 75
meters
of optical
fiber.[10]
[
edit]Gravitational time dilation tests
In
1959 Ro
bert
Pound a
nd Glen
A.
Rebka m
easured
the very
slight gra
vitational
red
shift in
the
frequenc
y of light
emitted
at a
lower
height,
where
Earth's
gravitatio
nal field
is
relatively
more
intense.
The
results
were
within
10% of
the
predictio
ns of
general
relativity.
Later
Pound
and
Snider
(in 1964)
derived
an even
closer
result of
1%. This
effect is
as
predicte
d by
gravitatio
nal time
dilation.
(See Po
und–
Rebka
experim
ent)
In 2010
gravitatio
nal time
dilation
was
measure
d at the
Earth's
surface
with a
height
differenc
e of only
one
meter,
using
optical
atomic
clocks.[10]
[
edit]Velocity and gravitational time
dilation combined-effect tests
Hafele
and
Keating,
in 1971,
flew cae
sium ato
mic
clocks
east and
west
around
the Earth
in
commer
cial
airliners,
to
compare
the
elapsed
time
against
that of a
clock
that
remaine
d at
the US
Naval
Observat
ory. Two
opposite
effects
came
into play.
The
clocks
were
expected
to age
more
quickly
(show a
larger
elapsed
time)
than the
referenc
e clock,
since
they
were in a
higher
(weaker)
gravitatio
nal
potential
for most
of the
trip
(c.f. Pou
nd,
Rebka).
But also,
contrasti
ngly, the
moving
clocks
were
expected
to age
more
slowly
because
of the
speed of
their
travel.
From the
actual
flight
paths of
each
trip, the
theory
predicte
d that
the flying
clocks,
compare
d with
referenc
e clocks
at the
U.S.
Naval
Observat
ory,
should
have lost
40+/-23
nanosec
onds
during
the
eastward
trip and
should
have
gained
275+/-21
nanosec
onds
during
the
westwar
d trip.
Relative
to the
atomic
time
scale of
the U.S.
Naval
Observat
ory, the
flying
clocks
lost
59+/-10
nanosec
onds
during
the
eastward
trip and
gained
273+/-7
nanosec
onds
during
the
westwar
d trip
(where
the error
bars
represen
t
standard
deviation
). [11]In
2005,
the Natio
nal
Physical
Laborato
ry in
the Unite
d
Kingdom
reported
their
limited
replicatio
n of this
experim
ent.[12] T
he NPL
experim
ent
differed
from the
original
in that
the
caesium
clocks
were
sent on
a shorter
trip
(London
–
Washing
ton D.C.
return),
but the
clocks
were
more
accurate
. The
reported
results
are
within
4% of
the
predictio
ns of
relativity.
The Glo
bal
Positioni
ng
System
can be
consider
ed a
continuo
usly
operatin
g
experim
ent in
both
special
and
general
relativity.
The in-
orbit
clocks
are
correcte
d for
both
special
and
general
relativisti
c time
dilation
effects a
s
describe
d above,
so that
(as
observe
d from
the
Earth's
surface)
they run
at the
same
rate as
clocks
on the
surface
of the
Earth.
[edit]Muon lifetime
A
comparison
of muon lifet
imes at
different
speeds is
possible. In
the
laboratory,
slow muons
are
produced,
and in the
atmosphere
very fast
moving
muons are
introduced
by cosmic
rays. Taking
the muon
lifetime at
rest as the
laboratory
value of
2.22 μs, the
lifetime of a
cosmic ray
produced
muon
traveling at
98% of the
speed of
light is about
five times
longer, in
agreement
with
observation
s.[13] In this
experiment
the "clock"
is the time
taken by
processes
leading to
muon
decay, and
these
processes
take place in
the moving
muon at its
own "clock
rate", which
is much
slower than
the
laboratory
clock.
[edit]Time dilation and space flight
Time
dilation
would make
it possible
for
passengers
in a fast-
moving
vehicle to
travel further
into the
future while
aging very
little, in that
their great
speed slows
down the
rate of
passage of
on-board
time. That
is, the ship's
clock (and
according to
relativity,
any human
travelling
with it)
shows less
elapsed
time than
the clocks of
observers
on Earth.
For
sufficiently
high speeds
the effect is
dramatic.
For
example,
one year of
travel might
correspond
to ten years
at home.
Indeed, a
constant
1 g accelera
tion would
permit
humans to
travel
through the
entire
known
Universe in
one human
lifetime.[14] T
he space
travellers
could return
to Earth
billions of
years in the
future. A
scenario
based on
this idea
was
presented in
the
novel Planet
of the
Apes by Pie
rre Boulle.
A more
likely use of
this effect
would be to
enable
humans to
travel to
nearby stars
without
spending
their entire
lives aboard
the ship.
However,
any such
application
of time
dilation
during inters
tellar
travel would
require the
use of some
new,
advanced
method
of propulsio
n. The Orion
Project has
been the
only major
attempt
toward this
idea.
Current
space flight
technology
has
fundamental
theoretical
limits based
on the
practical
problem that
an
increasing
amount of
energy is
required for
propulsion
as a craft
approaches
the speed of
light. The
likelihood of
collision with
small space
debris and
other
particulate
material is
another
practical
limitation. At
the
velocities
presently
attained,
however,
time dilation
is not a
factor in
space
travel.
Travel to
regions of
space-time
where
gravitational
time dilation
is taking
place, such
as within the
gravitational
field of a
black hole
but outside
theevent
horizon (per
haps on a
hyperbolic
trajectory
exiting the
field), could
also yield
results
consistent
with present
theory.
[edit]Time dilation at constant acceleration
In special
relativity,
time dilation
is most
simply
described in
circumstanc
es where
relative
velocity is
unchanging.
Nevertheles
s, the
Lorentz
equations
allow one to
calculate pr
oper
time and
movement
in space for
the simple
case of a
spaceship
whose accel
eration,
relative to
some
referent
object in
uniform (i.e.
constant
velocity)
motion,
equals g thr
oughout the
period of
measureme
nt.
Let t be the
time in an
inertial
frame
subsequentl
y called the
rest frame.
Let x be a
spatial
coordinate,
and let the
direction of
the constant
acceleration
as well as
the
spaceship's
velocity
(relative to
the rest
frame) be
parallel to
the x-axis.
Assuming
the
spaceship's
position at
time t = 0
being x = 0
and the
velocity
being v0 and
defining the
following
abbreviation
the
following
formulas
hold:[15]
Position:
Velo
city:
P
r
o
p
e
r
ti
m
e
:
In
th
e
ca
se
w
he
re
v(
0)
=
v0
=
0
an
d τ
(0)
=
τ0
=
0 t
he
int
eg
ral
ca
n
be
ex
pr
es
se
d
as
a
lo
ga
rit
h
mi
c
fu
nc
tio
n
or,
eq
ui
va
le
ntl
y,
as
an
in
ve
rs
e
hy
pe
rb
oli
c
fu
nc
tio
n:
[
edit]Spacetime geometry of velocity time dilation
Time dilation
in transverse
motion
The
green
dots
and
red
dots
in the
anima
tion
repres
ent
space
ships.
The
ships
of the
green
fleet
have
no
velocit
y
relativ
e to
each
other,
so for
the
clocks
onboa
rd the
individ
ual
ships
the
same
amou
nt of
time
elaps
es
relativ
e to
each
other,
and
they
can
set up
a
proce
dure
to
maint
ain a
synch
ronize
d
stand
ard
fleet
time.
The
ships
of the
"red
fleet"
are
movin
g with
a
velocit
y of
0.866
of the
speed
of
light
with
respe
ct to
the
green
fleet.
The
blue
dots
repres
ent
pulses
of
light.
One
cycle
of
light-
pulses
betwe
en
two
green
ships
takes
two
secon
ds of
"green
time",
one
secon
d for
each
leg.
As
seen
from
the
persp
ective
of the
reds,
the
transit
time
of the
light
pulses
they
excha
nge
amon
g
each
other
is one
secon
d of
"red
time"
for
each
leg.
As
seen
from
the
persp
ective
of the
green
s, the
red
ships'
cycle
of
excha
nging
light
pulses
travel
s a
diago
nal
path
that is
two
light-
secon
ds
long.
(As
seen
from
the
green
persp
ective
the
reds
travel
1.73 (
)
light-
secon
ds of
distan
ce for
every
two
secon
ds of
green
time.)
One
of the
red
ships
emits
a light
pulse
towar
ds the
green
s
every
secon
d of
red
time.
These
pulses
are
receiv
ed by
ships
of the
green
fleet
with
two-
secon
d
interv
als as
meas
ured
in
green
time.
Not
shown
in the
anima
tion is
that
all
aspec
ts of
physic
s are
propor
tionall
y
involv
ed.
The
light
pulses
that
are
emitte
d by
the
reds
at a
partic
ular
freque
ncy as
meas
ured
in red
time
are
receiv
ed at
a
lower
freque
ncy as
meas
ured
by the
detect
ors of
the
green
fleet
that
meas
ure
again
st
green
time,
and
vice
versa.
The
anima
tion
cycles
betwe
en the
green
persp
ective
and
the
red
persp
ective,
to
emph
asize
the
symm
etry.
As
there
is no
such
thing
as
absol
ute
motio
n in
relativi
ty (as
is also
the
case
for Ne
wtonia
n
mech
anics)
, both
the
green
and
the
red
fleet
are
entitle
d to
consid
er
thems
elves
motio
nless i
n their
own
frame
of
refere
nce.
Again,
it is
vital to
under
stand
that
the
result
s of
these
intera
ctions
and
calcul
ations
reflect
the
real
state
of the
ships
as it
emerg
es
from
their
situati
on of
relativ
e
motio
n. It is
not a
mere
quirk
of the
metho
d of
meas
ureme
nt or
comm
unicati
on.
[edit]